Ch2 Mathematical models of systems Introduction Models in time-domain (Differential equations) Models in frequency-domain(Transfer function) Block diagram models Signal-flow graph model Simulation and Examples

2.1 Introduction Mathematical models can help us to understand and control complex system effectively. So it is necessary to analyze the relationships between the system variables and to obtain a mathematical model. Because the systems are dynamic in nature, the descriptive equations are usually differential equations, if they can be linearized, then Laplace transform can be used to simplify the method of solution.

Goals of modeling Modeling method Analysis Design or Control Analytical method Experimental method

Models under Consideration Linear Time-invariant (constant) Parameter-lumped

2.2 Differential equations of physical systems Through-variable and across-variable Analogous variables Refer to (P34-37) Voltage-Velocity analogy Force-current analogy

How to obtain DE ? Newton’s laws for Mechanical system Kirchhoff’s laws for electrical systems Two examples (Refer to script 2-2)

Solving the equation Analytical or classical method Laplace transform method Example of solving differential equation ( Refer to script 2-3,4)

Modes of dynamic system Modes is determined by Characteristic roots: real and distinct roots real and repeated roots complex conjugate roots

2.3 Linear approximations of physical systems Non-linearity is essential and pervasive A system is defined as linear in terms of the system excitation and response. Principle of superposition Homogeneity A linear system satisfies the properties of superposition and homogeneity

Linear approximations method Taylor series expansion Example: pendulum oscillator model Refer to P40

2.4 The Laplace transform Review complex algebraic (refer to P41-47): Inverse Laplace transform residue evaluation Final value theorem

Time response by Laplace transform Time response solution is obtained by: 1. obtain the differential equations 2. obtain the Laplace transformation of differential equations 3. solve the algebraic equation of the variable of interest 4. obtain the response by the inverse Laplace transform

2.5 The transfer function of linear system Transfer function is defined as the ratio of the Laplace transform of the output variable to the laplace transform of the input variable , with all initial conditions assumed to be zero. Examples: (refer to P48)

Characteristics of TF （传递函数的特性） Black-box model rational real fractional function transition between TF and DE TF and impulse response Transfer function is the Laplace transform of the impulse response and vice versa.

Implication of TF （传递函数的含义） Zero initial condition representation of poles and zeros Inputs and outputs of system are zero at t=0-

TF of typical elements and dynamic systems Basic factors of TF Examples of TF Refer to P50-57 and Table 2.5 (P58-61)

2.6 Block diagram models （结构图或方框图模型） Block diagrams representation is prevalent in control engineering, and consist of unidirectional operational blocks that represent the transfer function of the variables of interests. Representation of block diagram Refer to P62-63

Transformation of Block Diagram BD=Scheme+Equation Four Components of BD (refer to 2-14,15) Transition between TF and BD Refer to 2-15,16,17

Characteristics of BD Unidirectional Principle Without load effect Connection ways: Series parallel and feedback connection

Block diagram reduction Combination of blocks in cascade Combination of blocks in parallel Elimination of feedback loop Moving pickoff points and summing points Principle: Keep every signal to be equivalent

Examples of BD reduction Refer to textbook ( P64-66) Other examples : refer to (2-19,20,21)

Assignment (课后作业） Review Ch2 (P32--) E2.4 E2.6 E2.8 E2.18